Software - SpectraPLUS-RT seems good but I'd like a cheaper or free alternative, preferably that works cross platform.

That's a toy, not likely to be useful for taking good measurements. For instance the only mention I can see of an FFT window function only mentions the Hanning window which is not used for measuring. You need to select a flattop window for accurate measurement of signal peaks.

Also it doesn't seem to mention noise measurements, which using an FFT have to be done differently to the spectral peak measurements as both the window function, sampling rate and number of points in the FFT affect the noise floor seen in a plot. For noise measurement the requirements of an FFT window are different.

I wrote some analysis tools in Python using the scipy.signal library, so for instance I save a .WAV from audacity, then post process for various things, often using FFT but not exclusively, for instance measure SiNaD or just distortion. Something I can do (not actually using the FFT it happens) is determine the test tone freq/amp/phase by refining a model iteratively, then subtract it from the data to plot the residual distortion+noise, which can be very revealing.

So when looking for some software be on the lookout for a tool that knows about spectral peak measurement using flattop window(s), and separate noise measurements, and ideally distortion-residual plotting.

Useful reading about measuring with an FFT:

https://holometer.fnal.gov/GH_FFT.pdf